Problem $7.$ An L$-$shape body is suspended in vertical a$)$ Derive the EoM for $$ using FBD and FLD$.$

b$)$ Calculate the equilibrium angle ${}^{*}$ numerically.

$($Hint: use the fsolve command in Matlab.$)$

c$)$ Linearize the EoM about the equilibrium. Calculate

the natural angular frequency of the linearized system.

Solve the linearized EoM for $(0)=0$ and ${}^{}(0)=$

${}_0=1[ra\frac{d}{s}]$ analytically.

d$)$ Simulate the motion in Matlab by solving the

nonlinear EoM numerically. Plot the solution $(t)$

over tin$[0,5]s.$ Plot the analytical solution $(t)$ of

the linearized EoM in the same graph. Submit the

figure and your codes.

plane as shown in the figure. The body consists of two

bars of length $l=0\mathrm{.}5m$ and mass $m=3kg,$ while

the suspension is two springs of stiffness $k=100$

$\frac{N}{m}.$ Assume that the springs remain horizontal and

vertical. Use $J_O=\frac{2}{3}m{l}^2$ and $g=9\mathrm{.}81\frac{m}{s^2}.$